{
  "id": 197,
  "curve_key": "760704094801311927921:-21402602511704044518245286166281",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-15848001975027331832",
    "24771530688804644612096284539"
  ],
  "rank_lower_bound": 14,
  "naive_height": 144.2821306343028,
  "faltings_height": 9.91865462217029,
  "conductor": "158660341771519260190542491906863061372490",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "17",
    "131",
    "6811583",
    "117316757",
    "141514035854442770479"
  ],
  "discriminant": "-10343870453933375939366089310754943399947700700823152640000",
  "regulator": "91751195039.7385955277149593413022755814531860544813979568131122369",
  "points": [
    [
      "-64560819529/25",
      "27521684792689179/125"
    ],
    [
      "36123117509/25",
      "8740159303518117/125"
    ],
    [
      "12682101068781/4225",
      "17895139115611159401/274625"
    ],
    [
      "272865715950557/167281",
      "3906800984102749854129/68417929"
    ],
    [
      "53671483231/9",
      "10186605986437651/27"
    ],
    [
      "428560504493/49",
      "255573047630702943/343"
    ],
    [
      "12449957567",
      "1325597777680281"
    ],
    [
      "15851815757",
      "1938237977452041"
    ],
    [
      "897117523463477/44521",
      "26382156112824679385091/9393931"
    ],
    [
      "7586604431813/169",
      "20816879163190357437/2197"
    ],
    [
      "-1901953423",
      "219166447716621"
    ],
    [
      "-313569788467/289",
      "991035603369385113/4913"
    ],
    [
      "-2964269157/4",
      "1520192145743473/8"
    ],
    [
      "2016208521369317/885481",
      "18467414160521026345821/833237621"
    ]
  ],
  "submitter": "Seewoo Lee",
  "commentary": "Rank ≥14. Mestre–Fermigier construction from the integer 6-tuple a=[-138, 138, 162, -60, -90, -12] with shift t=1227/5: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 14 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.",
  "created_at": "2026-06-25 16:03:29",
  "updated_at": "2026-07-01 22:43:40"
}